1
$\begingroup$

I am trying to solve the equation of the following type: $$u''(x) = (a-b\operatorname{sech}^2(cx))u(x),$$ where $a,b,c$ are positive real constants.

It is my first time when I met the equation which includes $\operatorname{sech}^2(x)u$ function.

Any hint will be good for me! Thank you in advance!

  • 0
    @Moo yeah, i saw that, but I need to solve it anyway :/ – 2017-02-09
  • 0
    @Moo no, i need exact form of the solution. I am not sure if it solvable, but this article https://pdfs.semanticscholar.org/4725/6fa58ee1fd9ef8b9eb71814ef46dc5a12d56.pdf (Example 3.1.1) says that the particular case can be solved by hypergeometric series. – 2017-02-09
  • 0
    It is a pitty that $c\neq 1$. In such a case, there is an explicit solution in terms of associated Legendre polynomials. – 2017-02-09
  • 0
    @ClaudeLeibovici this equation I got from the spectral problem for the operator $L_+$ of the NLS equation with nonlinearity of type $|u|^{2p}u$. The number $c$ in the equation is defined by $c=p$, so it is possible that $c \neq 1$ :/ – 2017-02-09

0 Answers 0